In this paper we study in detail a family of continued fraction expansions of any number in the unit closed interval [0,1] whose digits are differences of consecutive non-positive integer powers of an integer m⩾2. For the transformation which generates this expansion and its invariant measure, the Perron–Frobenius operator is given and studied. For this expansion, we apply the method of random systems with complete connections by Iosifescu and

We show a correspondence between simple continued fraction expansions of irrational numbers and irreducible permutative representations of the Cuntz algebra O_∞. In regard to the correspondence, it is shown that the equivalence of real numbers with respect to modular transformations is equivalent to the unitary equivalence of representations. Furthermore, we show that quadratic irrationals are related to irreducible permutative representations